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These statements are commonly referred to as **Thales' Theorem** and the converse of Thales' Theorem.

Consider a right triangle $ABC,$ with $m∠B=90_{∘},$ inscribed in a circle.

According to the Inscribed Angle Theorem, the measure of $∠B$ is half the measure of its intercepted arc, $AC.$

Substituting $m∠B=90_{∘}$ into the equation above gives that $mAC=180_{∘}.$ Then, $AC$ is a semicircle, implying that $AC$ $($the hypotenuse of $△ABC)$ is a diameter of the circle.

Consider a triangle $ABC$ inscribed in a circle such that one side of the triangle is a diameter of the circle.

Since $AC$ is a diameter, then $AC$ is a semicircle and then $mAC=180_{∘}.$ Now, the$m∠B =21 mAC=21 (180_{∘})=90_{∘} $

This implies that $∠B$ is a right angle, which makes $△ABC$ a right triangle.